The Draining Hourglass · AI Generator #GEN-28CA717A

Physics · fluid-mechanics

Problem

An hourglass contains fine sand that flows through a circular opening of diameter d=2.0  mmd = 2.0\;\text{mm} at the bottom of the upper chamber. The upper chamber has a cylindrical shape with cross-sectional area A=15  cm2A = 15\;\text{cm}^2. When the sand level is at height h=8.0  cmh = 8.0\;\text{cm} above the opening, the sand flows out at a rate of Q=0.50  cm3/sQ = 0.50\;\text{cm}^3/\text{s}. The sand grains have density ρs=2650  kg/m3\rho_s = 2650\;\text{kg/m}^3 and the bulk density of the sand (accounting for air gaps) is ρb=1500  kg/m3\rho_b = 1500\;\text{kg/m}^3. The effective viscosity of the flowing sand can be modeled as η=0.15  Pa⋅s\eta = 0.15\;\text{Pa}\cdot\text{s}. (a) Find the velocity of sand flow through the opening when h=8.0  cmh = 8.0\;\text{cm}. (b) If the sand were to flow like an ideal fluid under gravity alone (following Torricelli's law), what would be the flow velocity through the opening? (c) Determine the effective pressure drop across the opening due to viscous resistance of the sand flow.

Related theory

Browse the problem library